91Ó°ÊÓ

Let \(G\) be a finite group. Suppose that the only automorphism of \(G\) is the identity. Prove that \(G\) is abelian and that every element has order 2 [After you know the structure theorem for abelian groups, or after you have read the general definition of a vector space, and viewing \(G\) as vector space over Z 27 , prove that \(G\) has at most 2 elements.]

(a) Show that the commutator subgroup is a normal subgroup. (b) Show that \(G G^{e}\) is abelian.

(a) Let \(G\) be a solvable group, and \(H\) a subgroup. Prove that \(H\) is solvable. (b) Let \(G\) be a solvable group, and \(f: G \rightarrow G\) a surjective homomorphism. Show that \(G^{\prime}\) is solvable.